310
Buoyancy Forced Circulation and Cross-Gyre Flow
In particular, we may calculate v2 directly:
V 2 =
Y2
8h = l_
WE
< O
JR cos e 8¢ f3h (1 + r 12 <1> 2 )
(5.4.13)
since the Ekman pumping is negative in the subtropical gyre. Thus the meridional velocity in layer 2 is always southward in the ventilated zone regardless
of the strength of the heating or cooling. The magnitude of the heating and
cooling can affect the magnitude of the velocity and the extent of the domain of
the ventilated solution, but within that domain the meridional velocity must be
negative, and therefore the vertical velocity in layer 2 at the interface between
layers 1 and 2 must also be negative. Therefore, if the layer is heated, so that
w. > 0, the circulation in the ventilated region must be indirect, as we found in
the numerical solution of Luyten and Stommel (1986b) in the previous section.
It follows here directly from the form of the ventilated solution (5.4.1).
The streamline which separates the subducted fluid from the recirculating
fluid driven by the heating can be found by the same process as in Section 4.4.
We examine the path of the streamline which leaves the outcrop line in its most
eastern position along the eastern boundary where, at that position, D6 = 0
and h = H 2 . Thus, as before, we obtain as the parametric equation for the
eastern boundary of the ventilated region:
(5.4.14)
It is important to recall that the boundary determined by (5.4.14) is not the
shadow zone boundary determined by the characteristic emanating from the
intersection of the outcrop line and the eastern boundary. This boundary lies to
the east of the eastern edge of the ventilated zone, as was shown in Fig. 5.3.5.
The zone in between the two curves is the domain of the southward return flow
of the buoyancy-driven cyclonic circulation which is wedged between the
ventilated zone and the eastern boundary.
Generally, for a specified ratio of the cross-isopycnal velocity to Ekman
velocity, b(f), (5.4.10) must be integrated numerically, starting at ( = 1 and
proceeding southward towards ( = 0. It is somewhat more natural to consider
an interval in which the dependent variable increases, and the change of
variable:
!1=1-(
(5.4.15)
allows (5.4.10) to be written:
d
2
(1- 11) d 11
= (1- <1>)- b(1 + r 12<1> )
(5.4.16)
over the interval 0 :::; 11 :::; 1, with the initial condition <1>(11 = 0) = 0.
When there is no cross-isopycnal flux, and b = 0, the solution of (5.4.16)
which satisfies the initial condition is:
Buoyancy Forced Circulation and Cross-Gyre Flow
In particular, we may calculate v2 directly:
V 2 =
Y2
8h = l_
WE
< O
JR cos e 8¢ f3h (1 + r 12 <1> 2 )
(5.4.13)
since the Ekman pumping is negative in the subtropical gyre. Thus the meridional velocity in layer 2 is always southward in the ventilated zone regardless
of the strength of the heating or cooling. The magnitude of the heating and
cooling can affect the magnitude of the velocity and the extent of the domain of
the ventilated solution, but within that domain the meridional velocity must be
negative, and therefore the vertical velocity in layer 2 at the interface between
layers 1 and 2 must also be negative. Therefore, if the layer is heated, so that
w. > 0, the circulation in the ventilated region must be indirect, as we found in
the numerical solution of Luyten and Stommel (1986b) in the previous section.
It follows here directly from the form of the ventilated solution (5.4.1).
The streamline which separates the subducted fluid from the recirculating
fluid driven by the heating can be found by the same process as in Section 4.4.
We examine the path of the streamline which leaves the outcrop line in its most
eastern position along the eastern boundary where, at that position, D6 = 0
and h = H 2 . Thus, as before, we obtain as the parametric equation for the
eastern boundary of the ventilated region:
(5.4.14)
It is important to recall that the boundary determined by (5.4.14) is not the
shadow zone boundary determined by the characteristic emanating from the
intersection of the outcrop line and the eastern boundary. This boundary lies to
the east of the eastern edge of the ventilated zone, as was shown in Fig. 5.3.5.
The zone in between the two curves is the domain of the southward return flow
of the buoyancy-driven cyclonic circulation which is wedged between the
ventilated zone and the eastern boundary.
Generally, for a specified ratio of the cross-isopycnal velocity to Ekman
velocity, b(f), (5.4.10) must be integrated numerically, starting at ( = 1 and
proceeding southward towards ( = 0. It is somewhat more natural to consider
an interval in which the dependent variable increases, and the change of
variable:
!1=1-(
(5.4.15)
allows (5.4.10) to be written:
d
2
(1- 11) d 11
= (1- <1>)- b(1 + r 12<1> )
(5.4.16)
over the interval 0 :::; 11 :::; 1, with the initial condition <1>(11 = 0) = 0.
When there is no cross-isopycnal flux, and b = 0, the solution of (5.4.16)
which satisfies the initial condition is: